I am so happy that you are considering taking time to go to Estonia. Hopefully you love it there. Old Town Tallinn is incredible. And Estonia is easy for English speakers to get around since all the signs are in three languages, Estonian, Russian, and English. Most people speak English since it has been taught in school in Estonia from the third grade since 1990.
Tallinn airport is very close to Old Town Tallinn. There is a tram that can bring you from the airport to most parts of town. Last I heard all public transit in Tallinn is free.
I recommend getting the Tallinn Card which is a bus and tram pass, discount card, and it gives you access to many museums for free. https://www.visittallinn.ee/eng/tallinncard
Walking around old town is a wonderful thing to do in sunny weather. Beyond that, here is a brief list of things worth seeing in Tallinn.
For sunny weather
- Don't miss Toompea, or upper town. That is where parliament, government offices, and a lot of embassies are. There are fewer tourist things to see, but it is worth exploring. Don't miss out on seeing the city from one of the two viewing platforms
- Kohtuotsa https://goo.gl/maps/GeEHVDHf6KoxDe728
- Patkuli https://goo.gl/maps/cQDeSdW84rfGjWBB6
- Kadriorg Park - This is where the Presidential House (Kadriorg Palace) is, as well as a Japanese garden, and many walking paths within the well cared for garden. There is a very good art museum there too. You can take the city tram there.
- Take a walk North East along the coast and explore the Maarjamäe Palace - Estonian History Museum. In the same complex is a film and music museum.
- If you like getting to the top of things, the Tallinn TV Tower has a foot dangling walk on the outside of the tower. Loads of fun.
Rainy weather
- Seaplane Harbour, Estonian Maritime Museum is a large and wonderful place to explore. This is a must see.
- To see what life was like in the middle ages, Kiek in de Kök Museum and Bastion Tunnels.
- Lunch or dinner at a historical theme restaurant Peppersack, Olde Hansa, or III Draakon.
A great bar with uniquely crafted drinks is the Sigmund Freud Bar. https://goo.gl/maps/UceMuvqbLjxit5P69
If you like the 80s hit band Depeche Mode, the Depeche Mode Baar is a great place to visit. https://maps.app.goo.gl/f1rKP2WUMiA7yLLJ7
A hole in the wall German restaurant. Restaurant Munga Kelder. https://maps.app.goo.gl/EQBxnTA9NG2Xs41PA
A cool place to hang out and go out to eat is the Telliskivi Creative City. It is a short walk NW of Old Town on the other side of the train station. There is also a great market at the train station. This is an area where locals hang out. https://maps.app.goo.gl/eJPXiYny8DEixcyD7
Here are my favorite pictures from my trips to Estonia https://k7try.com/photos/2017-18%20Estonia/
Tallinn is a wonderful place to visit. If you want to explore more, you can take the ferry to Helsinki from Tallinn to further extend your adventures.